Dear Uwe, there are the notions of properad and prop. They are successive generalisations of operads: for operads the composable configurations are rooted trees; for properads the composable configurations are directed graphs required to be connected and acyclic; for props, connectedness is given up. Note that in all these cases, as well as in the string diagram calculus for monoidal categories, the edges are the objects, and the nodes are the operations. The (1,1)-operations are just arrows in the sense of categories. This is in contrast with the role graphs play as structures underlying categories. Properads can be described as algebras for a monad on a category of presheaves on elementary graphs. Any (directed) hypergraph defines a presheaf on elementary graphs, and the properad monad actually restricts to this subcategory: there is a monad for properads defined on the category of hypergraphs. The category of directed graphs embeds in two ways into hypergraphs: one sending edges to hyperedges, and another sending edges to (1,1)-nodes (and nodes to hyperedges). Under this second embedding, the free-properad monad restricts to the free-category monad! In this way hypergraphs provide the smallest setting in which to unify the dual roles played by graphs (as contrasted above), and in this way I think one can say that properads provide an answer to your question (unless I misunderstand it). All this is explained in my recent paper 'Graphs, hypergraphs, and properads', arXiv:1407.3744. Cheers, Joachim. On 25/6/15 14:34, Uwe Egbert Wolter wrote:

Dear all,

Categories can be defined by extending directed multi graphs with identities and composition. The underlying directed multi graph of a small category C is given by the sets C_Mor, C_Obj and the source and target maps src^C,trg^C:C_Mor -> C_Obj.

Has it ever been investigated what structures arise when we try, instead, to extend directed multi hypergraphs by identities and composition? A "directed multi hypergraph" H is thereby given by a set H_E of edges, a set H_V of vertices and two maps src^H,trg^H:H_E -> Pow(H_V) from H_E into the power set Pow(H_V) of H_V.

I'm aware of monoidal categories and I would like to know if there is something else around.

Any comment or reference is welcome

Uwe Wolter

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